Optimal recovery of isotropic classes of rth differentiable multivariate functions

Estimates are given for the optimal recovery of functions in d variables, which are known to have (r?1)st absolutely continuous and rth bounded derivatives in any direction over, either a bounded convex d-dimensional body G, or which are periodic over a d dimensional lattice. The information is the values of the function and all its derivatives of order less than r at n points. We obtain some asymptotic estimates for this problem, and some exact results for several special cases which contain the results of Babenko, Borodachov, and Skorokhodov.     

Optimal recovery on the classes of functions with bounded mixed derivative

Temlyakov considered the optimal recovery on the classes of functions with bounded mixed derivative in the Lp metrics and gave the upper estimates of the optimal recovery errors. In this paper, we determine the asymptotic orders of the optimal recovery in Sobolev spaces by standard information, i.e., function values, and give the nearly optimal algorithms which attain the asymptotic orders of the optimal recovery.     

Optimal Recovery of Isotropic Classes of rth Differentiable Functions on ℝ d

We consider the optimal recovery problem of isotropic classes of r-th differentiable multivariate functions defined on ? ^d , and obtain some asymptotically optimal results. It turns out that this optimal recovery problem is intimately related to the optimal covering problem of ? ^d by equal balls in discrete geometry.     

On the construction of optimal cubature formulae which use integrals over hyperspheres

We consider formulae of approximate integration over a d$d$-dimensional ball which use n$n$ surface integrals along (d-1)$(d-1)$-dimensional spheres centered at the origin. For a class of functions defined on the ball with gradients satisfying an integral restriction, optimal formulae of this type are obtained.     

Optimal recovery of integral operators and its applications

In this paper we present the solution to a problem of recovering a rather arbitrary integral operator based on incomplete information with error. We apply the main result to obtain optimal methods of recovery and compute the optimal error for the solutions to certain integral equations as well as boundary and initial value problems for various PDE’s.     

Optimal cubature formulas for tensor products of certain classes of functions

We study the problem of constructing an optimal formula of approximate integration along a d-dimensional parallelepiped. Our construction utilizes mean values along intersections of the integration domain with n hyperplanes of dimension (d−1), each of which is perpendicular to some coordinate axis. We find an optimal cubature formula of this type for two classes of functions. The first class controls the moduli of continuity with respect to all variables, whereas the second class is the intersection of certain periodic multivariate Sobolev classes. We prove that all node hyperplanes of the optimal formula in each case are perpendicular to a certain coordinate axis and are equally spaced and the weights are equal. For specific moduli of continuity and for sufficiently large n, the formula remains optimal for the first class among cubature formulas with arbitrary positions of hyperplanes.     

Optimal information for approximating periodic analytic functions

Let be a strip in the complex plane. For fixed integer let denote the class of -periodic functions , which are analytic in and satisfy in . Denote by the subset of functions from that are real-valued on the real axis. Given a function , we try to recover at a fixed point by an algorithm on the basis of the information

where , are the Fourier coefficients of . We find the intrinsic error of recovery

Furthermore the -dimensional optimal information error, optimal sampling error and -widths of in , the space of continuous functions on , are determined. The optimal sampling error turns out to be strictly greater than the optimal information error. Finally the same problems are investigated for the class , consisting of all -periodic functions, which are analytic in with -integrable boundary values. In the case sampling fails to yield optimal information as well in odd as in even dimensions.

     

Optimal recovery methods for the integral on classes of differentiable functions

Найдены методы восст ановления интеграла по информации $$I\left( f \right) = \left\{ {f^{(j)} \left( {x_i } \right)\left( {j = 0, ..., \gamma _i - 1; i = 1, ..., n; 1 \leqq \gamma _i \leqq r; \gamma _i + ... + \gamma _n \leqq N} \right.} \right\},$$ оптимальные на класс ахW _p ^r ,r=1,2,...; 1≦p≦∞. Это позволило, в частност и, получить наилучшие для классаW _p ^r квадратурные форму лы вида $$\mathop \smallint \limits_0^1 f\left( x \right)dx = \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 1}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right)$$ И $$\mathop \smallint \limits_0^1 f\left( x \right)dx = af\left( 0 \right) + \mathop \Sigma \limits_{i = 1}^n \mathop \Sigma \limits_{j = 0}^{\gamma _i - 1} a_{ij} f^{(j)} \left( {x_i } \right) + bf\left( 1 \right) + \mathop \Sigma \limits_{j = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} b_j f^{(2j - 1)} \left( 0 \right) + \mathop \Sigma \limits_{k = 1}^{[{r \mathord{\left/ {\vphantom {r 2}} \right. \kern-\nulldelimiterspace} 2}]} c_k f^{(2k - 1)} \left( 1 \right) + R\left( f \right).$$      

Optimal paths in bi-attribute networks with fractional cost functions

An important routing problem is to determine an optimal path through a multi-attribute network which minimizes a cost function of path attributes. In this paper, we study an optimal path problem in a bi-attribute network where the cost function for path evaluation is fractional. The problem can be equivalently formulated as the “bi-attribute rational path problem” which is known to be NP-complete. We develop an exact approach to find an optimal simple path through the network when arc attributes are non-negative. The approach uses some path preference structures and elimination techniques to discard, from further consideration, those (partial) paths that cannot be parts of an optimal path. Our extensive computational results demonstrate that the proposed method can find optimal paths for large networks in very attractive times.     

Optimal recovery of solutions of the generalized heat equation in the unit ball from inaccurate data

We consider the problem of optimal recovery of solutions of the generalized heat equation in the unit ball. Information is given at two time instances, but inaccurate. The solution is to be constructed at some intermediate time. We provide the optimal error and present an algorithm which achieves this error level.     

Decomposition and approximation of multivariate functions on the cube

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1]d is a finite sum of the form ∑jφjψj , where each φj can be extended to a smooth periodic function, each ψj is an algebraic polynomial, and each φjψj is a product of separated variable type and its smoothness is same as f . Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1]d can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error.     

Approximation of infinitely differentiable multivariate functions is intractable

We prove that L_∞-approximation of C^∞-functions defined on [0,1]^d is intractable and suffers from the curse of dimensionality. This is done by showing that the minimal number of linear functionals needed to obtain an algorithm with worst case error at most ε(0,1) is exponential in d. This holds despite the fact that the rate of convergence is infinite.     

A local duality principle for the Baire classes of functions

A local dual of a Banach space X is a closed subspace of X^∗ that satisfies the properties that the principle of local reflexivity assigns to X as a subspace of X∗∗. We show that, for every ordinal 1?α?ω₁, the spaces Bα[0,1] of bounded Baire functions of class α are local dual spaces of the space M[0,1] of all Borel measures. As a consequence, we derive that each annihilator Bα^⊥[0,1] is the kernel of a norm-one projection.     

各向异性Besov-Wiener类由多重样本的最优恢复

研究各向异性Besov-Wiener类SrpqθB（R^n）在Lq（R^n）,（1＜q≤P＜∞）中由其函数和它们的导数样本的最优恢复问题,确定了误差界的精确阶.     

Optimal lower bounds for cubature error on the sphere

We show that the worst-case cubature error E(Q_m;H^s) of an m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s=H^s(S²),s>1, has the lower bound , where the constant c_s is independent of Q_m and m. This lower bound result is optimal, since we have established in previous work that there exist sequences of cubature rules for which with a constant independent of n. The method of proof is constructive: given the cubature rule Q_m, we construct explicitly a ‘bad’ function f_mH^s, which is a function for which Q_mf_m=0 and . The construction uses results about packings of spherical caps on the sphere.     

Convolution properties for certain classes of multivalent functions

Recently N.E. Cho, O.S. Kwon and H.M. Srivastava [Nak Eun Cho, Oh Sang Kwon, H.M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators, J. Math. Anal. Appl. 292 (2004) 470-483] have introduced the class of multivalent analytic functions and have given a number of results. This class has been defined by means of a special linear operator associated with the Gaussian hypergeometric function. In this paper we have extended some of the previous results and have given other properties of this class. We have made use of differential subordinations and properties of convolution in geometric function theory.     

Approximation of Sobolev classes by polynomials and ridge functions

Let be the usual Sobolev class of functions on the unit ball in , and be the subclass of all radial functions in . We show that for the classes and , the orders of best approximation by polynomials in coincide. We also obtain exact orders of best approximation in of the classes by ridge functions and, as an immediate consequence, we obtain the same orders in for the usual Sobolev classes .     

Optimal control of sources on some classes of functions

In this work, we investigate the problems of the control of sources’ motion and power, which influence the state of the objects described by partial differential equations. The functions defining the sources’ operation are taken from different classes of functions that are easy to implement from the technical point of view. For numerical solution to the problems considered, we obtain and validate formulae for the gradient of a target functional, which allow using first-order optimization methods, e.g. gradient projection method.